`Sorting Pennies, A Wagering Activity for Algebra 1David Drymiller                Marie Sklodowska Curie Metro H.S.                               4959 S. Archer Ave.                               Chicago IL 60632                               (312) 535-2120Objectives:     To abstract the idea of a line.      To introduce other methods of solving simultaneous equation besides         graphing.      To develop problem solving skills.  Materials Needed:     A scale able to read (to the nearest tenth of a gram) up to 400 grams     at least 1,000 pennies     a plastic cup to hold pennies on the scale     numbered plastic ziplock bags 1 through 8     16 index cards, 2 per ziplock bag     Strategy:          The following instructions are directed to the instructor:     Place the 1,000 pennies in a large pile on the front desk along withthe scale, the plastic cup and the ziplock bags.  Divide the class into 8 groups.  From each group have one member come up and take a handful of pennies.Weigh and record the amount 'W'.  Place the weighed pennies in thecorresponding ziplock bag and give the group member the instructions toseparate the pennies into two groups: 1981 and before 'B', 1983 and after 'A'.They are to record the number of each kind of penny on one of the index cards provided and keep it from your view.  On the other index card, they are to writethe total number of pennies in the bag 'T'.  Place this second index card and all pennies in the plastic bag and return to front desk.  As the bags arereturned record the number of pennies and calculate the predicted number ofpennies for 1982 and before for each group.  When all the pennies have beenreturned and the calculations finished, announce that through the miracle ofalgebra you know how many of each type they have sorted with a margin of errorof plus or minus two pennies.  (I find a small wager, if possible, with eachgroup enlivens the activity).  Collect your winnings and explain to the classwhy you won.        Pennies for 1981 and before weigh 3.1 grams    Pennies for 1983 and after weigh 2.5 grams    (Note: avoid 1982 pennies because the mint produced some of each.)                         Number     Weight    Total Weight     1981 and before        B         3.1       3.1B    1983 and after         A         2.5       2.5A    before and after       T                    W    Which gives us the equations:    3.1B + 2.5A = W                                        B +    A = T                                           The problem becomes just finding the intersection of two lines.    Solving by Substitution:  A = T - B                                     3.1B + 2.5( T - B )  = W                              3.1B + 2.5T - 2.5B   = W                                          0.6B + 2.5T   = W                                          - 2.5T   = - 2.5T                                     0.6B          = W - 2.5T                                               B   = (W - 2.5T) / 0.6    I use this activity after I have taught the graphing of two lines and finding their intersection but before other methods of finding the solutionof systems of equations.  This is an introductory lesson that abstracts the intersection of two lines to find a solution.`